Mathematics 10 — Exercise 3.5 · Q8 (ii)

Full question

Verify (AB)⁻¹ = B⁻¹A⁻¹.

Step-by-step solution

1. Compute the left side first, following the bracket order.

2. 2 columns in the first matrix match 2 rows in the second. The product has 2 rows and 2 columns.

   $$(\left[\begin{array}{cc}1& 2\\ 8& 0\end{array}\right],\left[\begin{array}{cc}0& -1\\ 5& 2\end{array}\right])$$

3. Entry (1,1): pair row 1 with column 1 and add their products.

   $$1\times 0+2\times 5$$

4. Calculate this individual product, including its signs.

   $$1\times 0=0$$

5. Calculate this individual product, including its signs.

   $$2\times 5=10$$

6. Add the calculated products to obtain this entry.

   $$0+10=10$$

7. Entry (1,2): pair row 1 with column 2 and add their products.

   $$1\times \left(-1\right)+2\times 2$$

8. Calculate this individual product, including its signs.

   $$1\times \left(-1\right)=-1$$

9. Calculate this individual product, including its signs.

   $$2\times 2=4$$

10. Add the calculated products to obtain this entry.

    $$-1+4=3$$

11. Entry (2,1): pair row 2 with column 1 and add their products.

    $$8\times 0+0\times 5$$

12. Calculate this individual product, including its signs.

    $$8\times 0=0$$

13. Calculate this individual product, including its signs.

    $$0\times 5=0$$

14. Add the calculated products to obtain this entry.

    $$0+0=0$$

15. Entry (2,2): pair row 2 with column 2 and add their products.

    $$8\times \left(-1\right)+0\times 2$$

16. Calculate this individual product, including its signs.

    $$8\times \left(-1\right)=-8$$

17. Calculate this individual product, including its signs.

    $$0\times 2=0$$

18. Add the calculated products to obtain this entry.

    $$-8+0=-8$$

19. Place the calculated entries in their original row and column positions.

    $$\left[\begin{array}{cc}10& 3\\ 0& -8\end{array}\right]$$

20. For a 2 × 2 determinant, multiply the main diagonal and subtract the other diagonal product.

    $$D=10\times \left(-8\right)-3\times 0$$

21. Calculate the first diagonal product.

    $$10\times \left(-8\right)=-80$$

22. Calculate the second diagonal product.

    $$3\times 0=0$$

23. Subtract the second product, including its sign.

    $$D=-80+0$$

24. Simplify the subtraction.

    $$D=-80$$

25. The determinant is nonzero, so the inverse exists.

    $$-80\ne 0$$

26. Swap the two diagonal entries to begin the adjoint.

    $$\left[\begin{array}{cc}-8& 3\\ 0& 10\end{array}\right]$$

27. Change the signs of the two off-diagonal entries.

    $$\left[\begin{array}{cc}-8& -3\\ 0& 10\end{array}\right]$$

28. Calculate this operation before combining the remaining terms.

    $$\frac{1}{-80}=-\frac{1}{80}$$

29. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{\left(-8\right)\times \left(-1\right)}{1\times 80}$$

30. Calculate these two products separately.

    $$\frac{8}{80}$$

31. The numerator and denominator have this greatest common divisor. Divide BOTH by it.

    $$\mathrm{GCD}=8$$

32. Show the two divisions explicitly.

    $$\frac{\frac{8}{8}}{\frac{80}{8}}$$

33. Evaluate the numerator division.

    $$\frac{8}{8}=1$$

34. Evaluate the denominator division.

    $$\frac{80}{8}=10$$

35. Write the reduced fraction.

    $$\frac{1}{10}$$

36. Calculate this operation before combining the remaining terms.

    $$\left(-8\right)\times \left(-\frac{1}{80}\right)=\frac{1}{10}$$

37. Divide adjoint entry (1,1) by the determinant.

    $$-\frac{8}{-80}=\frac{1}{10}$$

38. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{\left(-3\right)\times \left(-1\right)}{1\times 80}$$

39. Calculate these two products separately.

    $$\frac{3}{80}$$

40. Calculate this operation before combining the remaining terms.

    $$\left(-3\right)\times \left(-\frac{1}{80}\right)=\frac{3}{80}$$

41. Divide adjoint entry (1,2) by the determinant.

    $$-\frac{3}{-80}=\frac{3}{80}$$

42. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{0\times \left(-1\right)}{1\times 80}$$

43. Calculate these two products separately.

    $$\frac{0}{80}$$

44. The numerator and denominator have this greatest common divisor. Divide BOTH by it.

    $$\mathrm{GCD}=80$$

45. Show the two divisions explicitly.

    $$\frac{\frac{0}{80}}{\frac{80}{80}}$$

46. Evaluate the numerator division.

    $$\frac{0}{80}=0$$

47. Evaluate the denominator division.

    $$\frac{80}{80}=1$$

48. Write the reduced fraction.

    $$0\times \frac{1}{1}$$

49. Calculate this operation before combining the remaining terms.

    $$0\times \left(-\frac{1}{80}\right)=0$$

50. Divide adjoint entry (2,1) by the determinant.

    $$\frac{0}{-80}=0$$

51. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{10\times \left(-1\right)}{1\times 80}$$

52. Calculate these two products separately.

    $$-\frac{10}{80}$$

53. The numerator and denominator have this greatest common divisor. Divide BOTH by it.

    $$\mathrm{GCD}=10$$

54. Show the two divisions explicitly.

    $$\frac{-\frac{10}{10}}{\frac{80}{10}}$$

55. Evaluate the numerator division.

    $$-\frac{10}{10}=-1$$

56. Evaluate the denominator division.

    $$\frac{80}{10}=8$$

57. Write the reduced fraction.

    $$-\frac{1}{8}$$

58. Calculate this operation before combining the remaining terms.

    $$10\times \left(-\frac{1}{80}\right)=-\frac{1}{8}$$

59. Divide adjoint entry (2,2) by the determinant.

    $$\frac{10}{-80}=-\frac{1}{8}$$

60. Place the calculated entries in their original row and column positions.

    $$\left[\begin{array}{cc}\frac{1}{10}& \frac{3}{80}\\ 0& -\frac{1}{8}\end{array}\right]$$

61. Now compute the right side independently.

62. For a 2 × 2 determinant, multiply the main diagonal and subtract the other diagonal product.

    $$D=0\times 2-\left(-1\right)\times 5$$

63. Calculate the first diagonal product.

    $$0\times 2=0$$

64. Calculate the second diagonal product.

    $$\left(-1\right)\times 5=-5$$

65. Subtract the second product, including its sign.

    $$D=0+5$$

66. Simplify the subtraction.

    $$D=5$$

67. The determinant is nonzero, so the inverse exists.

    $$5\ne 0$$

68. Swap the two diagonal entries to begin the adjoint.

    $$\left[\begin{array}{cc}2& -1\\ 5& 0\end{array}\right]$$

69. Change the signs of the two off-diagonal entries.

    $$\left[\begin{array}{cc}2& 1\\ -5& 0\end{array}\right]$$

70. Divide adjoint entry (1,1) by the determinant.

    $$\frac{2}{5}=\frac{2}{5}$$

71. Divide adjoint entry (1,2) by the determinant.

    $$\frac{1}{5}=\frac{1}{5}$$

72. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{\left(-5\right)\times 1}{1\times 5}$$

73. Calculate these two products separately.

    $$-\frac{5}{5}$$

74. The numerator and denominator have this greatest common divisor. Divide BOTH by it.

    $$\mathrm{GCD}=5$$

75. Show the two divisions explicitly.

    $$\frac{-\frac{5}{5}}{\frac{5}{5}}$$

76. Evaluate the denominator division.

    $$\frac{5}{5}=1$$

77. Write the reduced fraction.

    $$-\frac{1}{1}$$

78. Calculate this operation before combining the remaining terms.

    $$\left(-5\right)\times \frac{1}{5}=-1$$

79. Divide adjoint entry (2,1) by the determinant.

    $$-\frac{5}{5}=-1$$

80. Divide adjoint entry (2,2) by the determinant.

    $$\frac{0}{5}=0$$

81. Place the calculated entries in their original row and column positions.

    $$\left[\begin{array}{cc}\frac{2}{5}& \frac{1}{5}\\ -1& 0\end{array}\right]$$

82. For a 2 × 2 determinant, multiply the main diagonal and subtract the other diagonal product.

    $$D=1\times 0-2\times 8$$

83. Calculate the first diagonal product.

    $$1\times 0=0$$

84. Calculate the second diagonal product.

    $$2\times 8=16$$

85. Subtract the second product, including its sign.

    $$D=0-16$$

86. Simplify the subtraction.

    $$D=-16$$

87. The determinant is nonzero, so the inverse exists.

    $$-16\ne 0$$

88. Swap the two diagonal entries to begin the adjoint.

    $$\left[\begin{array}{cc}0& 2\\ 8& 1\end{array}\right]$$

89. Change the signs of the two off-diagonal entries.

    $$\left[\begin{array}{cc}0& -2\\ -8& 1\end{array}\right]$$

90. Calculate this operation before combining the remaining terms.

    $$\frac{1}{-16}=-\frac{1}{16}$$

91. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{0\times \left(-1\right)}{1\times 16}$$

92. Calculate these two products separately.

    $$\frac{0}{16}$$

93. The numerator and denominator have this greatest common divisor. Divide BOTH by it.

    $$\mathrm{GCD}=16$$

94. Show the two divisions explicitly.

    $$\frac{\frac{0}{16}}{\frac{16}{16}}$$

95. Evaluate the numerator division.

    $$\frac{0}{16}=0$$

96. Evaluate the denominator division.

    $$\frac{16}{16}=1$$

97. Divide adjoint entry (1,1) by the determinant.

    $$\frac{0}{-16}=0$$

98. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{\left(-2\right)\times \left(-1\right)}{1\times 16}$$

99. Calculate these two products separately.

    $$\frac{2}{16}$$

100. The numerator and denominator have this greatest common divisor. Divide BOTH by it.

     $$\mathrm{GCD}=2$$

101. Show the two divisions explicitly.

     $$\frac{\frac{2}{2}}{\frac{16}{2}}$$

102. Evaluate the numerator division.

     $$\frac{2}{2}=1$$

103. Evaluate the denominator division.

     $$\frac{16}{2}=8$$

104. Write the reduced fraction.

     $$\frac{1}{8}$$

105. Calculate this operation before combining the remaining terms.

     $$\left(-2\right)\times \left(-\frac{1}{16}\right)=\frac{1}{8}$$

106. Divide adjoint entry (1,2) by the determinant.

     $$-\frac{2}{-16}=\frac{1}{8}$$

107. For multiplication of fractions, multiply the numerators together and the denominators together.

     $$\frac{\left(-8\right)\times \left(-1\right)}{1\times 16}$$

108. Calculate these two products separately.

     $$\frac{8}{16}$$

109. Show the two divisions explicitly.

     $$\frac{\frac{8}{8}}{\frac{16}{8}}$$

110. Evaluate the denominator division.

     $$\frac{16}{8}=2$$

111. Write the reduced fraction.

     $$\frac{1}{2}$$

112. Calculate this operation before combining the remaining terms.

     $$\left(-8\right)\times \left(-\frac{1}{16}\right)=\frac{1}{2}$$

113. Divide adjoint entry (2,1) by the determinant.

     $$-\frac{8}{-16}=\frac{1}{2}$$

114. For multiplication of fractions, multiply the numerators together and the denominators together.

     $$\frac{1\times \left(-1\right)}{1\times 16}$$

115. Calculate these two products separately.

     $$-\frac{1}{16}$$

116. Calculate this operation before combining the remaining terms.

     $$1\times \left(-\frac{1}{16}\right)=-\frac{1}{16}$$

117. Divide adjoint entry (2,2) by the determinant.

     $$\frac{1}{-16}=-\frac{1}{16}$$

118. Place the calculated entries in their original row and column positions.

     $$\left[\begin{array}{cc}0& \frac{1}{8}\\ \frac{1}{2}& -\frac{1}{16}\end{array}\right]$$

119. 2 columns in the first matrix match 2 rows in the second. The product has 2 rows and 2 columns.

     $$(\left[\begin{array}{cc}\frac{2}{5}& \frac{1}{5}\\ -1& 0\end{array}\right],\left[\begin{array}{cc}0& \frac{1}{8}\\ \frac{1}{2}& -\frac{1}{16}\end{array}\right])$$

120. For multiplication of fractions, multiply the numerators together and the denominators together.

     $$\frac{2\times 1}{1\times 5}$$

121. Calculate these two products separately.

     $$\frac{2}{5}$$

122. Calculate this operation before combining the remaining terms.

     $$2\times \frac{1}{5}=\frac{2}{5}$$

123. For multiplication of fractions, multiply the numerators together and the denominators together.

     $$\frac{2\times 0}{5\times 1}$$

124. Calculate these two products separately.

     $$\frac{0}{5}$$

125. Show the two divisions explicitly.

     $$\frac{\frac{0}{5}}{\frac{5}{5}}$$

126. For multiplication of fractions, multiply the numerators together and the denominators together.

     $$\frac{1\times 1}{1\times 5}$$

127. Calculate these two products separately.

     $$\frac{1}{5}$$

128. Calculate this operation before combining the remaining terms.

     $$1\times \frac{1}{5}=\frac{1}{5}$$

129. For multiplication of fractions, multiply the numerators together and the denominators together.

     $$\frac{1\times 1}{1\times 2}$$

130. Calculate this operation before combining the remaining terms.

     $$1\times \frac{1}{2}=\frac{1}{2}$$

131. For multiplication of fractions, multiply the numerators together and the denominators together.

     $$\frac{1\times 1}{5\times 2}$$

132. To add these fractions, first use a common denominator: the least common multiple of their denominators.

     $$\mathrm{LCD}=10$$

133. Multiply this numerator AND denominator by the same factor. Its value stays unchanged.

     $$0\times \frac{1}{1}=\frac{0\times 10}{1\times 10}$$

134. Calculate the converted numerator; the denominator is now common.

     $$\frac{0}{10}$$

135. Multiply this numerator AND denominator by the same factor. Its value stays unchanged.

     $$\frac{1}{10}=\frac{1\times 1}{10\times 1}$$

136. Keep the common denominator and add the signed numerators.

     $$\frac{0+1}{10}=\frac{1}{10}$$

137. Entry (1,1): pair row 1 with column 1 and add their products.

     $$\frac{2}{5}\times 0+\frac{1}{5}\times \frac{1}{2}$$

138. Calculate this individual product, including its signs.

     $$\frac{2}{5}\times 0=0$$

139. Calculate this individual product, including its signs.

     $$\frac{1}{5}\times \frac{1}{2}=\frac{1}{10}$$

140. For multiplication of fractions, multiply the numerators together and the denominators together.

     $$\frac{1\times 1}{1\times 10}$$

141. Calculate this operation before combining the remaining terms.

     $$1\times \frac{1}{10}=\frac{1}{10}$$

142. Add the calculated products to obtain this entry.

     $$0+\frac{1}{10}=\frac{1}{10}$$

143. For multiplication of fractions, multiply the numerators together and the denominators together.

     $$\frac{1\times 1}{1\times 8}$$

144. Calculate this operation before combining the remaining terms.

     $$1\times \frac{1}{8}=\frac{1}{8}$$

145. For multiplication of fractions, multiply the numerators together and the denominators together.

     $$\frac{2\times 1}{5\times 8}$$

146. Calculate these two products separately.

     $$\frac{2}{40}$$

147. Show the two divisions explicitly.

     $$\frac{\frac{2}{2}}{\frac{40}{2}}$$

148. Evaluate the denominator division.

     $$\frac{40}{2}=20$$

149. Write the reduced fraction.

     $$\frac{1}{20}$$

150. For multiplication of fractions, multiply the numerators together and the denominators together.

     $$\frac{\left(-1\right)\times 1}{1\times 16}$$

151. Calculate this operation before combining the remaining terms.

     $$\left(-1\right)\times \frac{1}{16}=-\frac{1}{16}$$

152. For multiplication of fractions, multiply the numerators together and the denominators together.

     $$\frac{1\times \left(-1\right)}{5\times 16}$$

153. Calculate these two products separately.

     $$-\frac{1}{80}$$

154. To add these fractions, first use a common denominator: the least common multiple of their denominators.

     $$\mathrm{LCD}=80$$

155. Multiply this numerator AND denominator by the same factor. Its value stays unchanged.

     $$\frac{1}{20}=\frac{1\times 4}{20\times 4}$$

156. Calculate the converted numerator; the denominator is now common.

     $$\frac{4}{80}$$

157. Multiply this numerator AND denominator by the same factor. Its value stays unchanged.

     $$-\frac{1}{80}=\frac{\left(-1\right)\times 1}{80\times 1}$$

158. Keep the common denominator and add the signed numerators.

     $$\frac{4-1}{80}=\frac{3}{80}$$

159. Entry (1,2): pair row 1 with column 2 and add their products.

     $$\frac{2}{5}\times \frac{1}{8}+\frac{1}{5}\times \left(-\frac{1}{16}\right)$$

160. Calculate this individual product, including its signs.

     $$\frac{2}{5}\times \frac{1}{8}=\frac{1}{20}$$

161. Calculate this individual product, including its signs.

     $$\frac{1}{5}\times \left(-\frac{1}{16}\right)=-\frac{1}{80}$$

162. For multiplication of fractions, multiply the numerators together and the denominators together.

     $$\frac{1\times 1}{1\times 20}$$

163. Calculate this operation before combining the remaining terms.

     $$1\times \frac{1}{20}=\frac{1}{20}$$

164. For multiplication of fractions, multiply the numerators together and the denominators together.

     $$\frac{1\times 1}{1\times 80}$$

165. Calculate these two products separately.

     $$\frac{1}{80}$$

166. Calculate this operation before combining the remaining terms.

     $$1\times \frac{1}{80}=\frac{1}{80}$$

167. For multiplication of fractions, multiply the numerators together and the denominators together.

     $$\frac{\left(-1\right)\times 1}{1\times 80}$$

168. Calculate this operation before combining the remaining terms.

     $$\left(-1\right)\times \frac{1}{80}=-\frac{1}{80}$$

169. Add the calculated products to obtain this entry.

     $$\frac{1}{20}-\frac{1}{80}=\frac{3}{80}$$

170. For multiplication of fractions, multiply the numerators together and the denominators together.

     $$\frac{0\times 1}{1\times 2}$$

171. Calculate these two products separately.

     $$\frac{0}{2}$$

172. Show the two divisions explicitly.

     $$\frac{\frac{0}{2}}{\frac{2}{2}}$$

173. Evaluate the numerator division.

     $$\frac{0}{2}=0$$

174. Entry (2,1): pair row 2 with column 1 and add their products.

     $$-0+0\times \frac{1}{2}$$

175. Calculate this individual product, including its signs.

     $$\left(-1\right)\times 0=0$$

176. Calculate this individual product, including its signs.

     $$0\times \frac{1}{2}=0$$

177. Add the calculated products to obtain this entry.

     $$0+0=0$$

178. For multiplication of fractions, multiply the numerators together and the denominators together.

     $$\frac{\left(-1\right)\times 1}{1\times 8}$$

179. To add these fractions, first use a common denominator: the least common multiple of their denominators.

     $$\mathrm{LCD}=8$$

180. Multiply this numerator AND denominator by the same factor. Its value stays unchanged.

     $$-\frac{1}{8}=\frac{\left(-1\right)\times 1}{8\times 1}$$

181. Multiply this numerator AND denominator by the same factor. Its value stays unchanged.

     $$0\times \frac{1}{1}=\frac{0\times 8}{1\times 8}$$

182. Calculate the converted numerator; the denominator is now common.

     $$\frac{0}{8}$$

183. Keep the common denominator and add the signed numerators.

     $$\frac{-1+0}{8}=-\frac{1}{8}$$

184. Entry (2,2): pair row 2 with column 2 and add their products.

     $$-\frac{1}{8}+0\times \left(-\frac{1}{16}\right)$$

185. Calculate this individual product, including its signs.

     $$\left(-1\right)\times \frac{1}{8}=-\frac{1}{8}$$

186. Calculate this individual product, including its signs.

     $$0\times \left(-\frac{1}{16}\right)=0$$

187. Add the calculated products to obtain this entry.

     $$-\frac{1}{8}+0=-\frac{1}{8}$$

188. Place the calculated entries in their original row and column positions.

     $$\left[\begin{array}{cc}\frac{1}{10}& \frac{3}{80}\\ 0& -\frac{1}{8}\end{array}\right]$$

189. Every corresponding entry is equal, so the stated identity holds.

     $$(\left[\begin{array}{cc}\frac{1}{10}& \frac{3}{80}\\ 0& -\frac{1}{8}\end{array}\right],\left[\begin{array}{cc}\frac{1}{10}& \frac{3}{80}\\ 0& -\frac{1}{8}\end{array}\right])$$

Answer

Every corresponding entry is equal, so the stated identity holds.